Problems Open and Otherwise

Topological dynamics

Periodic points

• Describe the set of periodic points or points with periodic itineraries in various settings (non-zero topological entropy, piecewise isometries,…)
• Count them (find a nice zeta function)

Symbolic dynamics

• describe its structure (e.g., by defining an abstract class to which the considered symbolic dynamics belong and which implies the considered properties)
• relate it precisely to the PA dynamics (e.g., in which case does it have the same topological entropy?)

Entropy

• dependence on the PA dynamics
• computation from geometric and combinatorial data

Ergodic theory

SRB measures

• Simplify Tsujii's general construction
• Find something like it for the piecewise hyperbolic situation, maybe using Baladi-Gouëzel recent work (prove first a generic result?)
• Prove Palis conjecture on SRB measures for PA dynamics, for instance for the Lozi family

Maximal entropy measures

• Find in which settings:
  • they always exist (e.g., homeomorphisms in higher dimension?)
  • they form a finite-dimensional simplex